# P11166: Algebraicity of a Thue-Morse infinite product

- ID: `P11166`
- Reference: `algebraicity-of-thue-morse-product-q`
- Page: https://theoremdb.org/statements/P11166
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: b5a83bd9bdbf7dfdc7134c15b7360f889389e7bc
- Current Markdown: https://api.theoremdb.org/v1/statements/algebraicity-of-thue-morse-product-q?representation=markdown
- Record maturity: Reviewed problem

## The problem

For each integer \(n\ge 0\), let \(t(n)\in\{0,1\}\) be the parity of the number of 1s in the binary expansion of \(n\), and set \(u(n)=(-1)^{t(n)}\). The convergent product \[Q=\prod_{n=1}^{\infty}\left(\frac{2n}{2n+1}\right)^{u(n)}=1.6281601297189172488\ldots\] is called a Thue-Morse infinite product. Determine whether \(Q\) is algebraic over \(\mathbb{Q}\).

## Status

The reviewed record remains open.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `algebraicity-of-thue-morse-product-q`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.